We present an algorithm for computer verification of the
global structure of structurally stable planar vector fields.
Constructing analytical proofs for the qualitative properties of phase
portraits has been difficult. We try to avoid this barrier by
augmenting numerical computations of trajectories of dynamical systems
with error estimates that yield rigorous proofs. Our approach
lends itself to high-precision estimates, because the proofs are
broken into independent calculations whose length in floating-point
operations does not increase with increasing precision. The algorithm
is tested on a system that arises in the study of Hopf
bifurcation of periodic orbits with 1:4 resonance.